Question 1

Which of the following best describes the concept of a 'limit' in calculus, as applied to modeling population growth in biology?

Accepted Answer

The value that a function approaches as the input approaches a certain point.

Question 2

Suppose the cost $$C(x) to $$produce $$x$$ units of a product is given by $$C(x) = 500 + 20x. $$What is the marginal cost when $$x = 100$$?

Accepted Answer

$$20$$

Question 3

A function $$f(x) is $$continuous at $$x = a if $$which of the following conditions is satisfied?

Accepted Answer

$$\lim_{x 	o a} f(x) = f(a)$$

Question 4

The derivative of a function at a point represents:

Accepted Answer

The instantaneous rate of change of the function at that point.

Question 5

Given $$f(x) = x^2$, what is the average rate of change of f$ from x = 2$ to x = 5$?

Accepted Answer

$$\frac{f(5) - f(2)}{5 - 2} = \frac{25 - 4}{3} = 7$$

Question 6

Which of the following is the correct application of the Chain Rule for differentiating $$y = \sin(3x^2$)$$?

Accepted Answer

$$y' = \cos(3x^2$) \cdot 6x$$

Question 7

A hospital tracks the number of patients $$P(t)$$ admitted per day, modeled by $$P(t) = 100e$$^{0.05t}$$, where t$ is days since the start of the year. What is the instantaneous rate of change of admissions on day t = 10$?

Accepted Answer

$$P'(10) = 100 \cdot 0.05 \cdot e$$^{0.05 \cdot 10}$$ = 5e$$^{0.5}$$

Question 8

If $$f(x) is $$differentiable on $$[a, b]$$, which theorem guarantees that there exists $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}$$?

Accepted Answer

The Mean Value Theorem

Question 9

A pharmaceutical company models the concentration $$C(t) of a $$drug in the bloodstream by $$C(t) = 50e$$^{-0.1t}$$. What is the rate at which the concentration is decreasing at t = 5$ hours?

Accepted Answer

$$C'(5) = -5e$$^{-0.5}$$

Question 10

Which of the following statements about the definite integral $$\int_a^b f(x) dx is $$correct in the context of calculating total revenue over a time period?

Accepted Answer

It represents the accumulated total of $$f(x)$$ from $$x = a to$$ $$x = b.$$

Question 11

A population of bacteria grows according to $$P(t) = 200e$$^{0.03t}$$. How long will it take for the population to double?

Accepted Answer

$$t = \frac{\ln(2)}{0.03}$$

Question 12

Which of the following is NOT a requirement for the application of the Mean Value Theorem to a function $$f(x) on$$ $$[a, b]$$?

Accepted Answer

$$f(x)$$ must be increasing on $$[a, b]$$

Question 13

If $$R(x) is $$the revenue from selling $$x$$ units and $$R'(x) = 15$$, what does $$R'(x)$$ represent in a business context?

Accepted Answer

The marginal revenue per unit sold.

Question 14

A company’s profit function is $$P(x) = -2x^2$ + 40x - 100. $$What is the number of units $$x$$ that maximizes profit?

Accepted Answer

$$x = 10$$

Question 15

Which of the following best describes the Fundamental Theorem of Calculus?

Accepted Answer

It links the concept of differentiation and integration.

Question 16

Suppose $$f(x) = 3x^4$ - 2x^2$ + x. $$What is $$f''(x)$$, the second derivative?

Accepted Answer

$$f''(x) = 36x^2$ - 4$$

Question 17

A biologist models the rate of change of a population by $$\frac{dP}{dt} = 0.2P. If$$ $$P(0) = 100$$, what is $$P(5)$$?

Accepted Answer

$$P(5) = 100e$$^{1}$$

Question 18

Which of the following is the correct interpretation of $$\$$int_0$$^8 v(t) dt if$$ $$v(t) is $$the velocity of a car in meters per second?

Accepted Answer

The total distance traveled by the car from $$t = 0 to$$ $$t = 8$$ seconds.

Question 19

If $$f(x) = \ln(x)$$, what is $$\frac{d}{dx}f(x)$$?

Accepted Answer

$$\frac{1}{x}$$

Business Calculus: Core Concepts, Structure, and Success Strategies

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